{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "4e230913",
   "metadata": {},
   "source": [
    "# 创建一个线性数据集"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "1991a2f6",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:43.726884Z",
     "start_time": "2022-01-13T05:53:43.607203Z"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "x = 2 * np.random.rand(100, 1)  # 返回100*1的数组，x在0到1服从均匀分布\n",
    "y = 4 + 3 * x + np.random.rand(100, 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "689855d9",
   "metadata": {},
   "source": [
    "设置图像显示"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "865f7e85",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:44.060035Z",
     "start_time": "2022-01-13T05:53:43.727881Z"
    }
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "%config InlineBackend.figure_format = \"svg\"\n",
    "plt.rcParams['font.sans-serif'] = ['Microsoft YaHei']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "ca6ea24c",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:44.187652Z",
     "start_time": "2022-01-13T05:53:44.060990Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.0, 2.1, 4.0, 11.0)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
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     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(x, y)\n",
    "plt.axis([0,2.1,4,11])  # 界定轴范围，[x0,x1,y0,y1]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b651a6fa",
   "metadata": {},
   "source": [
    "实际函数：$y = 4 + 3x + \\epsilon$\n",
    "\n",
    "$\\epsilon$ 服从正态分布，也称高斯噪声"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "657c5fbb",
   "metadata": {},
   "source": [
    "## 成本函数和公式法"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f651bac1",
   "metadata": {},
   "source": [
    "线性公式：$\\hat{y}=\\theta x$，这里用的是“点乘”，点乘要求矩阵同形，对应位置对应相乘\n",
    "\n",
    "注意：在机器学习中，默认采用“点乘法”而非“矩阵乘法”\n",
    "\n",
    "$\\theta=[\\theta_0,\\theta_1, ... ,\\theta_m]$，其中$\\theta_0$是截距项，其余为系数（权重）项\n",
    "\n",
    "$x=[x_0, x_1, ..., x_m]$，其中$X_0$始终为1，在定义时直接定义为1，因为$X_0$对应的是$\\theta_0$，截距项不能乘以其他系数\n",
    "\n",
    "成本函数，即算均方差，这里的$\\theta$和X，y都是向量形式\n",
    "\n",
    "$MSE(X,h_0)=MSE(\\theta)=\\frac{1}{m}\\sum_{k=1}^m(\\theta^Tx^{(i)}-y^{(i)})^2$\n",
    "\n",
    "标准方程直接求参数，即公式法\n",
    "\n",
    "$\\hat{\\theta}=(X^TX)^{-1}X^Ty$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "34a34c7a",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:44.203643Z",
     "start_time": "2022-01-13T05:53:44.189645Z"
    }
   },
   "outputs": [],
   "source": [
    "x_b = np.c_[np.ones((100,1)), x]  # 每一个向量的 x0 都赋值为1，返回一个100*2的x矩阵"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "938b942d",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:44.219565Z",
     "start_time": "2022-01-13T05:53:44.204607Z"
    }
   },
   "outputs": [],
   "source": [
    "theta_best = np.linalg.inv(x_b.T.dot(x_b)).dot(x_b.T).dot(y)  # np.linalg.inv()求逆矩阵，这里是公式法"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "7d3097d3",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:44.234525Z",
     "start_time": "2022-01-13T05:53:44.221562Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[4.54642368],\n",
       "       [2.97157545]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "theta_best"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "24f4ba56",
   "metadata": {},
   "source": [
    "实际函数：$y = 4 + 3x + \\epsilon$\n",
    "\n",
    "可见非常接近"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "656a08e1",
   "metadata": {},
   "source": [
    "## 利用求得的参数预测"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "e7e11a7b",
   "metadata": {
    "ExecuteTime": {
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     "start_time": "2022-01-13T05:53:44.235525Z"
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   "outputs": [
    {
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      "text/plain": [
       "array([[ 4.54642368],\n",
       "       [10.48957457]])"
      ]
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     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
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    "x_new = np.array([[0],[2]])  # 预测矩阵x，2*1\n",
    "x_new_b = np.c_[np.ones((2,1)), x_new]  # 添加1到预测矩阵\n",
    "y_predict = x_new_b.dot(theta_best)  # x点乘之前得出的估计θ（theta_best）\n",
    "y_predict  # 这是预测的两个端点的值，相连就是预测线段"
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  },
  {
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   "execution_count": 8,
   "id": "d910e69e",
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     "start_time": "2022-01-13T05:53:44.252479Z"
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   ],
   "source": [
    "plt.plot(x_new, y_predict, 'r-', label='预测')\n",
    "plt.plot(x, y, 'b.')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a08afd04",
   "metadata": {},
   "source": [
    "### sklearn 方法"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "37b5b2e3",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:44.917700Z",
     "start_time": "2022-01-13T05:53:44.394100Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LinearRegression()"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sklearn.linear_model import LinearRegression\n",
    "lin_reg = LinearRegression()\n",
    "lin_reg.fit(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "63a4f684",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:44.933657Z",
     "start_time": "2022-01-13T05:53:44.920692Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([4.54642368]), array([[2.97157545]]))"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lin_reg.intercept_, lin_reg.coef_  # 截距项.intercept_，系数项.coef_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "3a930271",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:44.949615Z",
     "start_time": "2022-01-13T05:53:44.935652Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 4.54642368],\n",
       "       [10.48957457]])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lin_reg.predict(x_new)  # 预测值"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bd7e4c2b",
   "metadata": {},
   "source": [
    "### scipy（numpy）方法"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "861f6441",
   "metadata": {},
   "source": [
    "LinearRegression 基于scipy.linalg.lstsq()函数（最小二乘法）"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "2e8fb159",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:44.965610Z",
     "start_time": "2022-01-13T05:53:44.952606Z"
    }
   },
   "outputs": [],
   "source": [
    "theta_best_svd, residuals, rank, s = np.linalg.lstsq(x_b, y, rcond=1e-6)  # rcond 为误差截止值\n",
    "# theta_best_svd = np.linalg.lstsq(x_b, y, rcond=1e-6)[0]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2684ec4a",
   "metadata": {},
   "source": [
    "residuals 是残差，rank 是秩，一般只需要 theta_best_svd 即可"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "64392a63",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:44.981529Z",
     "start_time": "2022-01-13T05:53:44.966569Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[4.54642368],\n",
       "       [2.97157545]])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "theta_best_svd  # θ的预测向量"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "594f1bb3",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-09T05:41:21.642097Z",
     "start_time": "2022-01-09T05:41:21.626140Z"
    }
   },
   "source": [
    "np.linalg.lstsq 计算 $\\hat{\\theta}=X^{+}y$，$X^{+}$是X的伪逆矩阵，利用np.linalg.pinv()函数可以直接求伪逆\n",
    "\n",
    "一个与A的转置矩阵$A^T$同型的矩阵X，满足：$AXA=A，XAX=X$；此时，称矩阵X为矩阵A的伪逆，也称为广义逆矩阵。\n",
    "\n",
    "伪逆矩阵本身是用奇异值分解（SVD）的标准矩阵分解技术计算的，伪逆矩阵的求法不在这里讨论（毕竟我也不会），奇异矩阵（非方阵）的伪逆矩阵总是有定义的\n",
    "\n",
    "公式法的本质就是SVD，对于超大数据集，不宜采用公式法"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8a01fea6",
   "metadata": {},
   "source": [
    "### 复杂度"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5271da3a",
   "metadata": {},
   "source": [
    "标准方程计算$X^TX$的逆矩阵，$X^TX$是(n+1)×(n+1)型矩阵，其复杂度通常为$O(n^{2.4})$到$O(n^{3})$\n",
    "\n",
    "故，如果特征数量翻倍（要估计的参数×2），计算时间乘以大约 $2^{2.4}=5.3$ 到 $2^3=8$ 倍\n",
    "\n",
    "sklearn 的 linearRegression 类使用的SVD方法的复杂度约为$O(n^{2})$\n",
    "\n",
    "对于非超大型实例可以直接用SVD公式法，对于超大型实例，采用后面的方法"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8a9bdf23",
   "metadata": {},
   "source": [
    "# 梯度下降（Gradient Descent）"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d1fcc98e",
   "metadata": {},
   "source": [
    "## 批量梯度下降"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "59a6659f",
   "metadata": {},
   "source": [
    "梯度下降实际上是通过逐一求成本函数$MSE(\\theta_j)$的偏导数来确定下降方向\n",
    "\n",
    "批量梯度下降也称全梯度下降，是通过向量的形式一次性计算全部偏导数，得到一个向量\n",
    "\n",
    "* 成本函数：$MSE(X,h_0)=MSE(\\theta)=\\frac{1}{m}\\sum_{k=1}^m(\\theta^Tx^{(i)}-y^{(i)})^2$\n",
    "\n",
    "\n",
    "* 成本函数的偏导数：\n",
    "$\\frac{\\partial MSE(\\theta)}{\\partial\\theta_j}=\\frac{2}{m}\\sum_{k=1}^m(\\theta^Tx^{(i)}-y^{(i)})x_j^{(i)}$，即$\\theta^T$幂导再乘以系数$x_j^{(i)}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b8a8a996",
   "metadata": {},
   "source": [
    "梯度向量\n",
    "\n",
    "$\\nabla_{\\theta}MSE(\\theta)=[\\frac{\\partial}{\\partial\\theta_0}MSE(\\theta), \\quad\\frac{\\partial}{\\partial\\theta_1}MSE(\\theta),  \\quad...,  \\quad\\frac{\\partial}{\\partial\\theta_n}MSE(\\theta)]^T=\\frac{2}{m}X^T(X\\theta-y)$\n",
    "\n",
    "也就是上面的公式的向量表示形式\n",
    "\n",
    "算出梯度之后，如果梯度为正，即上坡，θ就减去$\\nabla_{\\theta}MSE(\\theta)×\\eta$:\n",
    "\n",
    "$\\theta^{(下一步)} \\quad =\\theta-\\eta\\nabla_{\\theta}MSE(\\theta)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "ddc11a81",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:44.997525Z",
     "start_time": "2022-01-13T05:53:44.983524Z"
    }
   },
   "outputs": [],
   "source": [
    "eta = 0.1  # 学习率等于0.1\n",
    "n_iteration = 1000  # 迭代次数\n",
    "m = 100  # 公式里面的分母m，即参数个数\n",
    "theta = np.random.randn(2,1)  # randn是标准正态分布，rand是[0,1)均匀分布，随机生成theta，自动下降到最优状态\n",
    "for _ in range(n_iteration):\n",
    "    gradients = 2/m * (x_b).T.dot(x_b.dot(theta) - y)  # 一次性全部计算出来\n",
    "    theta = theta - eta * gradients"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "71fb936f",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:45.013444Z",
     "start_time": "2022-01-13T05:53:44.998484Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[4.54642368],\n",
       "       [2.97157545]])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "theta"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7613745f",
   "metadata": {},
   "source": [
    "一般来说，要设置一个容差（范数）$\\epsilon$，即当梯度下降的值小于$\\epsilon$时结束迭代，节约时间"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9e60f9d9",
   "metadata": {},
   "source": [
    "## 随机梯度下降"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "eeefe2f4",
   "metadata": {},
   "source": [
    "利用模拟退火方法，逐步降低学习率，开始步长比较大，越往后越缩小，靠近全局最小值"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "96402131",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:45.093232Z",
     "start_time": "2022-01-13T05:53:45.015439Z"
    }
   },
   "outputs": [],
   "source": [
    "n_epochs = 50  # 迭代轮数\n",
    "t0, t1 = 5, 10  # 学习进度超参数，用于调整学习率\n",
    "\n",
    "def learning_schedule(t):\n",
    "    return t0 / (t + t1)\n",
    "\n",
    "theta = np.random.randn(2, 1)  # 随机确定θ初始值\n",
    "for epoch in range(n_epochs):\n",
    "    for i in range(m):  # m是实例个数，即在样本中抽100次值，每次单独计算一个，不需要批量计算\n",
    "        random_index = np.random.randint(m)  # 0到m随机取值\n",
    "        xi = x_b[random_index : random_index+1]\n",
    "        yi = y[random_index : random_index+1]\n",
    "        gradients = 2 * xi.T.dot(xi.dot(theta) - yi)\n",
    "        eta = learning_schedule(epoch * m + 1)\n",
    "        theta = theta - eta * gradients"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "e2802a40",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:45.109188Z",
     "start_time": "2022-01-13T05:53:45.094229Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[4.52569532],\n",
       "       [2.98319775]])"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "theta"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cec3de45",
   "metadata": {},
   "source": [
    "由于是随机抽的，有些样本可能从未被取到，如果要确保遍历全部实例，就要对训练集混洗（同步混洗特征和标签），然后逐个遍历，然后再次混洗，类推，收敛较慢。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a445d420",
   "metadata": {},
   "source": [
    "### SGDRegressor实现"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "d5e8deac",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:45.125145Z",
     "start_time": "2022-01-13T05:53:45.110187Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "SGDRegressor(eta0=0.1, penalty=None)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sklearn.linear_model import SGDRegressor\n",
    "sgd_reg = SGDRegressor(max_iter=1000, tol=1e-3, penalty=None, eta0=0.1)  \n",
    "# 设定最大迭代1000， 范数（tolerance）0.0003，学习率0.1，penalty表示正则化类型\n",
    "sgd_reg.fit(x, y.ravel())  # .ravel()是把多维数组转为一维数组，即全部合并到一个向量中，不加此方法会告警"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "dbbfe924",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:45.141105Z",
     "start_time": "2022-01-13T05:53:45.126144Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([4.5179913]), array([2.97921202]))"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sgd_reg.intercept_, sgd_reg.coef_  # 输出参数"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c627dc14",
   "metadata": {},
   "source": [
    "也比较接近"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3d3beefd",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-09T08:23:15.776877Z",
     "start_time": "2022-01-09T08:23:15.770854Z"
    }
   },
   "source": [
    "## 小批量梯度下降"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6dea6302",
   "metadata": {},
   "source": [
    "每次抽取一小批实例进行批量梯度下降，比一次性梯度下降性能好一点"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "da4daa9e",
   "metadata": {},
   "source": [
    "# 多项式回归"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "f266484f",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:45.251848Z",
     "start_time": "2022-01-13T05:53:45.142101Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
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     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    },
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       "  <style type=\"text/css\">*{stroke-linecap:butt;stroke-linejoin:round;}</style>\r\n",
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   "source": [
    "m = 100\n",
    "x = 6 * np.random.rand(m, 1) - 3  # np.random.rand(m, 1) 是0到1均匀分布的矩阵，x分布在 -3 ~ 3\n",
    "y = 0.5 * x**2 + x + 2 + np.random.randn(m, 1)  # 生成一批数据集\n",
    "plt.scatter(x, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7e48d8cf",
   "metadata": {},
   "source": [
    "## 多项式特征"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "76088ea3",
   "metadata": {},
   "source": [
    "$y = 0.5x^2 + x + 2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "51db7bad",
   "metadata": {
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     "end_time": "2022-01-13T05:53:45.267804Z",
     "start_time": "2022-01-13T05:53:45.254801Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.12012028, 0.01442888])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sklearn.preprocessing import PolynomialFeatures  # 多项式（Polynomia）特征\n",
    "poly_features = PolynomialFeatures(degree=2, include_bias=False)  # include_bias=False就不输出截距项系数1\n",
    "# degree设定需要拟合的最高次幂，该方法会自动添加所有特征（比较智能），但degree设定太高可能会数量爆炸\n",
    "x_poly = poly_features.fit_transform(x)\n",
    "x_poly[0]  # x_poly是转换后的x，现在包含原始特征和特征的平方，目的是增加一个新的特征，把x^2降为一阶的值"
   ]
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  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "1f97aa68",
   "metadata": {
    "ExecuteTime": {
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     "start_time": "2022-01-13T05:53:45.268763Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([1.70938366]), array([[0.9289607 , 0.53958493]]))"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "lin_reg = LinearRegression()\n",
    "lin_reg.fit(x_poly, y)\n",
    "lin_reg.intercept_, lin_reg.coef_  # 系数项的顺序是升幂顺序"
   ]
  },
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   "execution_count": 23,
   "id": "1a675ebc",
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     "start_time": "2022-01-13T05:53:45.284720Z"
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   "source": [
    "plt.scatter(x, y, label='原始数据')\n",
    "plt.scatter(x, lin_reg.predict(x_poly), label='预测数据')\n",
    "plt.legend()\n",
    "plt.show()"
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   "id": "ad13c249",
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  {
   "cell_type": "markdown",
   "id": "4bc15365",
   "metadata": {},
   "source": [
    "定义一个函数展示学习曲线"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "b8c439ca",
   "metadata": {
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     "end_time": "2022-01-13T05:53:45.505170Z",
     "start_time": "2022-01-13T05:53:45.490172Z"
    }
   },
   "outputs": [],
   "source": [
    "from sklearn.metrics import mean_squared_error\n",
    "from sklearn.model_selection import train_test_split\n",
    "\n",
    "def plot_learning_curves(model, x, y):\n",
    "    x_train, x_ver, y_train, y_ver = train_test_split(x, y, test_size=0.2)  # 划分训练集和验证集（verification）\n",
    "    train_errors, ver_errors = [], []\n",
    "    for m in range(1, len(x_train)):\n",
    "        model.fit(x_train[:m], y_train[:m])  # 训练实例随循环次数上升\n",
    "        y_train_predict = model.predict(x_train[:m])\n",
    "        y_ver_predict = model.predict(x_ver)  # 训练集无需切片，模型随训练集的扩大而改变，验证集的预测结果也改变了\n",
    "        train_errors.append(mean_squared_error(y_train[:m], y_train_predict))  # 记录在训练集上的标准差\n",
    "        ver_errors.append(mean_squared_error(y_ver, y_ver_predict))  # 记录验证集的标准差\n",
    "    plt.plot(np.sqrt(train_errors), 'r-+', linewidth=2, label='训练')\n",
    "    plt.plot(np.sqrt(ver_errors), 'b-', linewidth=3, label='验证')\n",
    "    plt.ylabel('RMSE')\n",
    "    plt.legend()"
   ]
  },
  {
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   "execution_count": 26,
   "id": "87d5d59b",
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     "start_time": "2022-01-13T05:53:45.506128Z"
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   },
   "outputs": [
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   "source": [
    "lin_reg = LinearRegression()\n",
    "plot_learning_curves(lin_reg, x, y)"
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  {
   "cell_type": "markdown",
   "id": "d1bb98c0",
   "metadata": {},
   "source": [
    "首先，误差都趋于1.5，如果最后趋于平稳的区域比较高，即RMSE没有逐渐到低位平缓，则存在欠拟合问题，其次，有时候验证集的误差反而低于训练集，其原因可能是数据太少，导致训练集和验证集的分布出现差别。\n",
    "\n",
    "出现欠拟合问题，增加更多数据无济于事（两误差之间已经很接近），应该采用更复杂的模型或提供更好的特征。"
   ]
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  {
   "cell_type": "markdown",
   "id": "0a4b4794",
   "metadata": {},
   "source": [
    "> 从置信区间角度理解标准差，标准差越小，说明预测值越精确"
   ]
  },
  {
   "cell_type": "code",
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   "id": "3afe4af4",
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     "start_time": "2022-01-13T05:53:45.729530Z"
    }
   },
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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
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     "output_type": "display_data"
    }
   ],
   "source": [
    "from sklearn.pipeline import Pipeline\n",
    "polynomial_regression = Pipeline([\n",
    "    ('ploy_feature', PolynomialFeatures(degree=10, include_bias=False)),\n",
    "    ('lin_reg', LinearRegression()),\n",
    "])\n",
    "\n",
    "plt.axis([0,80,0,3])  # 限定轴，否则该结果验证集误差过高，图像下方看不出差异\n",
    "plot_learning_curves(polynomial_regression, x, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2ccc7192",
   "metadata": {},
   "source": [
    "误差趋近1，比一次方程的误差1.5要好，其次训练集的效果比验证集要好，且间隔较大，说明存在过拟合，改善过拟合需要提高更多训练数据"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "911b94e3",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:46.251164Z",
     "start_time": "2022-01-13T05:53:46.001803Z"
    }
   },
   "outputs": [
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       "<Figure size 432x288 with 1 Axes>"
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     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from sklearn.pipeline import Pipeline\n",
    "polynomial_regression = Pipeline([\n",
    "    ('ploy_feature', PolynomialFeatures(degree=2, include_bias=False)),  # 改成二项式\n",
    "    ('lin_reg', LinearRegression()),\n",
    "])\n",
    "\n",
    "plt.axis([0,80,0,3])  # 限定轴，否则该结果验证集误差过高，图像下方看不出差异\n",
    "plot_learning_curves(polynomial_regression, x, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "76429b3c",
   "metadata": {},
   "source": [
    "显然二项式效果最好"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b77dd979",
   "metadata": {},
   "source": [
    "# 正则化线性模型"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7ee36a8c",
   "metadata": {},
   "source": [
    "## 岭回归 "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "14734ddc",
   "metadata": {},
   "source": [
    "岭回归之前最好先缩放数据，StandardScaler\n",
    "\n",
    "岭回归就是在成本函数中加上$\\alpha\\frac{1}{2}\\sum^{n}_{i=1}\\theta_i^2$，可以称之为惩罚函数，系数越大会导致成本增加\n",
    "\n",
    "* 成本函数 $J(\\theta)=MSE(\\theta)+\\alpha\\frac{1}{2}\\sum^{n}_{i=1}\\theta_i^2$\n",
    "\n",
    "注意，$\\theta$的角标从1开始，不包括截距项系数，$\\alpha$控制正则化程度，$\\alpha$增加则拟合曲线变平坦\n",
    "\n",
    "* 闭式解（公式法）：$\\hat{\\theta}=(X^TX+\\alpha A)^{-1}X^Ty$\n",
    "\n",
    "A是(n+1)×(n+1)对角方阵，左上角是0（除开截距项），对角线其余均为1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "2cb48a99",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:46.267095Z",
     "start_time": "2022-01-13T05:53:46.252136Z"
    }
   },
   "outputs": [],
   "source": [
    "np.random.seed(42)\n",
    "m = 20\n",
    "x = 3 * np.random.rand(m, 1)\n",
    "y = 1 + 0.5 * x + np.random.randn(m, 1) / 1.5\n",
    "x_new = np.linspace(0, 3, 100).reshape(100, 1)"
   ]
  },
  {
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   "execution_count": 30,
   "id": "5d4fa008",
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     "end_time": "2022-01-13T05:53:46.774777Z",
     "start_time": "2022-01-13T05:53:46.268093Z"
    }
   },
   "outputs": [
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       " <defs>\r\n",
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       "  </clipPath>\r\n",
       " </defs>\r\n",
       "</svg>\r\n"
      ],
      "text/plain": [
       "<Figure size 864x360 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from sklearn.preprocessing import StandardScaler\n",
    "from sklearn.linear_model import Ridge\n",
    "def plot_model(model_class, polynomial, alphas, **model_kargs):\n",
    "    for alpha, style in zip(alphas, (\"b-\", \"g--\", \"r:\")):\n",
    "        model = model_class(alpha, **model_kargs) if alpha > 0 else LinearRegression()\n",
    "        if polynomial:\n",
    "            model = Pipeline([\n",
    "                    (\"poly_features\", PolynomialFeatures(degree=10, include_bias=False)),\n",
    "                    (\"std_scaler\", StandardScaler()),\n",
    "                    (\"regul_reg\", model),\n",
    "                ])\n",
    "        model.fit(x, y)\n",
    "        y_new_regul = model.predict(x_new)\n",
    "        lw = 2 if alpha > 0 else 1\n",
    "        plt.plot(x_new, y_new_regul, style, linewidth=lw, label=r\"$\\alpha = {}$\".format(alpha))\n",
    "    plt.plot(x, y, \"b.\", linewidth=3)\n",
    "    plt.legend(loc=\"upper left\", fontsize=15)\n",
    "    plt.xlabel(\"$x_1$\", fontsize=18)\n",
    "    plt.axis([0, 3, 0, 4])\n",
    "\n",
    "plt.figure(figsize=(12,5))\n",
    "plt.subplot(121)\n",
    "plot_model(Ridge, polynomial=False, alphas=(0, 10, 100), random_state=42)\n",
    "plt.ylabel(\"$y$\", rotation=0, fontsize=18)\n",
    "plt.subplot(122)\n",
    "plot_model(Ridge, polynomial=True, alphas=(0, 10**-5, 1), random_state=42)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "382ab967",
   "metadata": {},
   "source": [
    "以上代码用于演示图像"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "63488126",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:46.790696Z",
     "start_time": "2022-01-13T05:53:46.776748Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1.55071465]])"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sklearn.linear_model import Ridge\n",
    "ridge_reg = Ridge(alpha=1, solver=\"cholesky\", random_state=42)\n",
    "ridge_reg.fit(x, y)\n",
    "ridge_reg.predict([[1.5]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "44043637",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:46.806692Z",
     "start_time": "2022-01-13T05:53:46.791694Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1.47012588])"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sgd_reg = SGDRegressor(penalty=\"l2\", max_iter=1000, tol=1e-3, random_state=42)\n",
    "# 梯度下降法，正则项类型为l2，即权重向量的l2范数平方的一半，也就是岭回归\n",
    "sgd_reg.fit(x, y.ravel())\n",
    "sgd_reg.predict([[1.5]])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4dea93a0",
   "metadata": {},
   "source": [
    "## Lasso回归"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e4336f10",
   "metadata": {},
   "source": [
    "* 成本函数 $J(\\theta)=MSE(\\theta)+\\alpha\\sum^{n}_{i=1}\\theta_i^2$\n",
    "\n",
    "与岭回归区别在于$\\sum^{n}_{i=1}\\theta_i^2$的系数是1，不是1/2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "25d260e4",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:46.822650Z",
     "start_time": "2022-01-13T05:53:46.807652Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1.47011206, 1.78031855])"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sgd_reg = SGDRegressor(penalty=\"l1\", max_iter=1000, tol=1e-3, random_state=42)\n",
    "# 梯度下降法，正则项类型为l1\n",
    "sgd_reg.fit(x, y.ravel())\n",
    "sgd_reg.predict([[1.5], [2]])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a6c674e3",
   "metadata": {},
   "source": [
    "## 弹性网络"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0d809ecc",
   "metadata": {},
   "source": [
    "$J(\\theta)=MSE(\\theta) + r\\alpha\\sum^{n}_{i=1}|\\theta_i|+\\frac{1-r}{2}\\alpha\\sum^{n}_{i=1}\\theta_i^2$\n",
    "\n",
    "r为混合比，该方法介于岭回归和Lasso回归之间"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "5eec05f5",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:46.838605Z",
     "start_time": "2022-01-13T05:53:46.823609Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1.54333232])"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sklearn.linear_model import ElasticNet\n",
    "elastic_net = ElasticNet(alpha=0.1, l1_ratio=0.5)  # l1_ratio 对应r\n",
    "elastic_net.fit(x,y)\n",
    "elastic_net.predict([[1.5]])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b0c0a082",
   "metadata": {},
   "source": [
    "有正则化比无正则化更好，默认岭回归，如果认为有效特征只有少数几个，应该倾向于使用Lasso或弹性网络，而更倾向于弹性网络"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "20e1a745",
   "metadata": {},
   "source": [
    "## 提前停止"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3c459b48",
   "metadata": {},
   "source": [
    "验证集误差达到最小时停止训练。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "a8d7a132",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:46.853527Z",
     "start_time": "2022-01-13T05:53:46.840562Z"
    }
   },
   "outputs": [],
   "source": [
    "np.random.seed(42)\n",
    "m = 100\n",
    "x = 6 * np.random.rand(m, 1) - 3\n",
    "y = 2 + x + 0.5 * x**2 + np.random.randn(m, 1)\n",
    "\n",
    "x_train, x_ver, y_train, y_ver = train_test_split(x[:50], y[:50].ravel(), test_size=0.5, random_state=10)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a78d20d0",
   "metadata": {},
   "source": [
    "基本实现方法如下："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "3fc90c52",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:47.504787Z",
     "start_time": "2022-01-13T05:53:46.854524Z"
    }
   },
   "outputs": [],
   "source": [
    "from sklearn.base import clone\n",
    "poly_scaler = Pipeline([\n",
    "    ('poly_features', PolynomialFeatures(degree=90, include_bias=False)),\n",
    "    ('std_scaler', StandardScaler()),\n",
    "])\n",
    "x_train_poly_scaled = poly_scaler.fit_transform(x_train)\n",
    "x_ver_poly_scaled = poly_scaler.transform(x_ver)  # 不是fit_transform\n",
    "sgd_reg = SGDRegressor(max_iter=1, tol=-np.infty, warm_start=True,\n",
    "                      penalty=None, learning_rate='constant', eta0=0.0005)\n",
    "                        # warm_start=True 时，下次在停止的地方继续训练\n",
    "minimum_val_error = float('inf')\n",
    "best_epoch = None\n",
    "best_model = None\n",
    "for epoch in range(1000):\n",
    "    sgd_reg.fit(x_train_poly_scaled, y_train)\n",
    "    y_ver_prdict = sgd_reg.predict(x_ver_poly_scaled)\n",
    "    val_error = mean_squared_error(y_ver, y_ver_prdict)\n",
    "    if val_error < minimum_val_error:  # 一旦误差升高就停止\n",
    "        minimun_val_error = val_error\n",
    "        best_epoch = epoch\n",
    "        best_model = clone(sgd_reg)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "6d0af4f9",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:47.520792Z",
     "start_time": "2022-01-13T05:53:47.506786Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "999"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "best_epoch"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "01210fd9",
   "metadata": {},
   "source": [
    "# 逻辑回归"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a1a120a4",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-11T08:52:52.412488Z",
     "start_time": "2022-01-11T08:52:52.388552Z"
    }
   },
   "source": [
    "逻辑回归计算输入特征的加权和（包括截距项），但输出结果的数理逻辑值而非输出结果\n",
    "\n",
    "逻辑回归模型的估计概率（向量形式）：\n",
    "\n",
    "$\\hat{p}=h_\\theta(x)=\\sigma(x^T\\theta)$\n",
    "\n",
    "其中 $\\sigma(t) = \\frac{1}{1+e^{-1}}$，为sigmoid函数\n",
    "\n",
    "逻辑回归返回将线性回归结果映射到sigmoid函数上的值（概率）"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "fc9465c1",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:47.648467Z",
     "start_time": "2022-01-13T05:53:47.522785Z"
    }
   },
   "outputs": [
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      ],
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(-10,10,1000)\n",
    "y = 1/(1+np.e**(-x))\n",
    "plt.plot(x,y,label='sigmoid 函数')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3deb73b1",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-11T08:57:00.077729Z",
     "start_time": "2022-01-11T08:57:00.061731Z"
    }
   },
   "source": [
    "$\n",
    "\\hat{y}=\\begin{cases}0，\\hat{p}<0.5 \\\\ 1，\\hat{p}\\geq0.5 \\end{cases}\n",
    "$\n",
    "\n",
    "$当x<0时，\\sigma(x)<0.5，当x\\geq0 时，\\sigma(x) \\geq 0.5，故如果x^T\\theta是正类，逻辑回归预测结果是1，负类为0$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "72e02578",
   "metadata": {},
   "source": [
    "## 训练和成本函数"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "89311c7c",
   "metadata": {},
   "source": [
    "单个训练实例的成本函数\n",
    "\n",
    "$\n",
    "c(\\theta)=\\begin{cases}  -ln(\\hat{p})，y = 1 \\\\ -ln(1-\\hat{p})，y = 0 \\end{cases}\n",
    "$\n",
    "\n",
    "y是实例，若对于正例，模型估计其概率接近0时，成本c会很高，同理若估计一个负例的概率接近1，成本c也会很高；概率接近1时，对于正例来说成本接近0，对于负例来说，成本会很高。\n",
    "\n",
    "> 所以预测正例概率越大，成本越低；预测负例概率越小，成本越低。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "542f633b",
   "metadata": {},
   "source": [
    "* 逻辑回归的成本函数\n",
    "\n",
    "$\n",
    "J(\\theta)=-\\frac{1}{m}[y^{(i)}ln(\\hat{p}^{(i)})+(1-y^{(i)})ln(1-\\hat{p}^{(i)})]\n",
    "$\n",
    "\n",
    "* 逻辑成本函数偏导数（梯度下降法）\n",
    "\n",
    "$\n",
    "\\frac{\\partial J(\\theta)}{\\partial\\theta_j} = \\frac{1}{m}\\sum^{m}_{i=1}[\\sigma(\\theta^Tx^{(i)})-y^{(i)}]x^{(i)}_j\n",
    "$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dbecbdac",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-11T09:34:19.820969Z",
     "start_time": "2022-01-11T09:34:19.809998Z"
    }
   },
   "source": [
    "## 决策边界"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "c1ffe87e",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:47.696316Z",
     "start_time": "2022-01-13T05:53:47.649401Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['data',\n",
       " 'target',\n",
       " 'frame',\n",
       " 'target_names',\n",
       " 'DESCR',\n",
       " 'feature_names',\n",
       " 'filename']"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 鸢尾花数据集\n",
    "from sklearn import datasets\n",
    "iris = datasets.load_iris()\n",
    "list(iris.keys())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "f752055d",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:47.712233Z",
     "start_time": "2022-01-13T05:53:47.697273Z"
    }
   },
   "outputs": [],
   "source": [
    "x = iris['data'][:, 3:]  # 这里仅研究花瓣宽度\n",
    "y = (iris['target'] == 2).astype(np.int32)  # 鸢尾花（正例）标记为1，否则0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "2ea6c461",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:47.728225Z",
     "start_time": "2022-01-13T05:53:47.713231Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LogisticRegression()"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sklearn.linear_model import LogisticRegression\n",
    "log_reg = LogisticRegression()\n",
    "log_reg.fit(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "6f4ea376",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:47.744185Z",
     "start_time": "2022-01-13T05:53:47.729189Z"
    }
   },
   "outputs": [],
   "source": [
    "x_new = np.linspace(0, 3, 1000).reshape(-1, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "637dcb27",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T05:53:47.760135Z",
     "start_time": "2022-01-13T05:53:47.745175Z"
    }
   },
   "outputs": [],
   "source": [
    "y_proba = log_reg.predict_proba(x_new)  # 返回的是概率"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "602a4120",
   "metadata": {
    "ExecuteTime": {
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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
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    }
   ],
   "source": [
    "plt.plot(x_new, y_proba[:, 1], 'g-', label='鸢尾花')\n",
    "plt.plot(x_new, y_proba[:, 0], 'b--', label='非鸢尾花')\n",
    "plt.xlabel('花瓣长度')\n",
    "plt.ylabel('概率')\n",
    "plt.legend()\n",
    "plt.plot([1.66, 1.66], [0, 1],'k:')  # [x1,x2], [y1, y2]\n",
    "plt.text(1.7, 0, '决策边界')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "id": "d8910a78",
   "metadata": {
    "ExecuteTime": {
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     "start_time": "2022-01-13T05:53:47.888761Z"
    }
   },
   "outputs": [
    {
     "data": {
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       "array([1, 0])"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "log_reg.predict([[1.7], [1.5]])  # 对于花瓣长度1.7和1.5的预测"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8b862c3a",
   "metadata": {},
   "source": [
    "## Softmax回归"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "79d01d11",
   "metadata": {
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     "end_time": "2022-01-13T04:26:57.309038Z",
     "start_time": "2022-01-13T04:26:57.286095Z"
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   "source": [
    "* Softmax回归分类器一次只能预测一个类（可以区分多类，但是一次只能输出某个类）\n",
    "\n",
    "1. 多元逻辑回归  \n",
    "\n",
    "    类k的Softmax分数：$ s_k(x) = x^T\\theta^{(k)} $，形式上与线性回归预测方程一样，返回多个类的分数\n",
    "    \n",
    "    每一个$\\theta^{(k)}$向量作为行储存在参数矩阵$\\Theta$中\n",
    "    \n",
    "    $Softmax$函数：$ \\hat{p}_k = \\sigma(s(x)_k) = \\frac{e^{s_{k}(x)}}{\\sum^{K}_{j}  e^{s_j(x)}} $，除以所有指数的和进行归一化\n",
    "\n",
    "2. 实际上：\n",
    "\n",
    "    $Sigmoid:\\hat{p}(x) = \\frac{1}{1+e^{-x}}$\n",
    "\n",
    "    $Softmax:\\hat{p}(x_1) = \\frac{e^{x_1}}{e^{x_1}+e^{x_2}+...+e^{x_n}} = \\frac{1}{1+e^{-(x_1+x_2+...+x_n)}} $\n",
    "\n",
    "3. Softmax回归分类预测：\n",
    "\n",
    "    $\\hat{y} = arg_k max[\\sigma(s(x))_k] = arg_k maxs_k(x) = arg_k max[(\\theta^{(k)})^Tx]$\n",
    "\n",
    "    $argmax$返回使函数最大化的变量值，在此式中，返回使得估计概率$\\sigma(s(x))_k$最大的$k$值\n",
    "    \n",
    "    \n",
    "4. 交叉熵成本函数\n",
    "    \n",
    "    $J(\\Theta)=-\\frac{1}{m}\\sum^m_{i=1}\\sum^K_{k=1}y^{(i)}log_2(\\hat{p}_k^{(i)})$\n",
    "    \n",
    "    $y_k^{(i)}$是属于类k的第i个实例的目标概率，一般而言等于0或1\n",
    "    \n",
    "    \n",
    "5. 交叉熵梯度向量\n",
    "    \n",
    "    $\\nabla_{\\theta(k)}J(\\Theta)=\\frac{1}{m}\\sum^m_{i=1}(\\hat{p}^{(i)}_k-y_k^{(1)})x^{(i)}$\n",
    "* 可以通过计算每个类的梯度向量，然后使用梯度下降，找到最小化成本函数的参数矩阵$\\Theta$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "id": "6fc24ca3",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T06:25:28.568237Z",
     "start_time": "2022-01-13T06:25:28.560260Z"
    }
   },
   "outputs": [],
   "source": [
    "x = iris['data'][:, (2, 3)]  # 花瓣长度，花瓣宽度\n",
    "y = iris['target']  # 包含三类花，用0，1，2表示"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "id": "11c94449",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T06:26:25.362360Z",
     "start_time": "2022-01-13T06:26:24.960333Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LogisticRegression(C=10, multi_class='multinomial')"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "softmax_reg = LogisticRegression(multi_class='multinomial', solver='lbfgs', C=10)\n",
    "softmax_reg.fit(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "id": "930a1b73",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T06:28:16.476815Z",
     "start_time": "2022-01-13T06:28:16.460861Z"
    }
   },
   "outputs": [
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       "array([2])"
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     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
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   "source": [
    "softmax_reg.predict([[5, 2]])"
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   "cell_type": "code",
   "execution_count": 50,
   "id": "0305faf8",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-01-13T06:28:32.740364Z",
     "start_time": "2022-01-13T06:28:32.736375Z"
    }
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   "outputs": [
    {
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       "array([[6.38014896e-07, 5.74929995e-02, 9.42506362e-01]])"
      ]
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     "execution_count": 50,
     "metadata": {},
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   "source": [
    "softmax_reg.predict_proba([[5,2]])"
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